Spectral radius and size conditions for fractional $(a,b,m)$-covered graphs

TL;DR

The paper addresses when a graph is fractional $(a,b,m)$-covered, generalizing fractional $[a,b]$-covered graphs by incorporating a fixed subgraph of $m$ edges. It develops two main criteria: a spectral-radius condition, requiring $n$ large enough, minimum degree at least $a+m$, and $ ho(G) \ge n-b-1$; and a size condition, requiring a corresponding edge-count bound. Both criteria apply to all integers $2 \le a \le b$ and $1 \le m \le b$, with specializations to the case $a=b=k$, and recover known results when $m=1$. The results extend fractional factor theory by providing explicit, comparable thresholds and highlight improvements over previous spectral bounds in related work.

Abstract

A fractional $(a,b,m)$-covered graph is a generalization of the concept of a fractional $[a,b]$-covered graph. For any $H \subseteq G$ with edge set $|E(H)| = m$, if there exists a fractional $[a,b]$-factor (the corresponding fractional indicator function is $h$) such that $h(e) = 1$ for any $e \in H$, then the graph $G$ is called a fractional $(a,b,m)$-covered graph. In this paper, we characterize the conditions for a graph to be a fractional $(a,b,m)$-covered graph from the perspectives of spectral radius and size, respectively.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3